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This paper considers the effect of additive white noise on the normal formfor the supercritical Hopf bifurcation in 2 dimensions. The main resultsinvolve the asymptotic behavior of the top Lyapunov exponent lambda associatedwith this random dynamical system as one or more of the parameters in thesystem tend to 0 or infinity. This enables the construction of a bifurcationdiagram in parameter space showing stable regions where lambda is negative(implying synchronization) and unstable regions where lambda is positive(implying chaotic behavior). The value of lambda depends strongly on theshearing effect of the twist factor of the deterministic Hopf bifurcation. Ifthe twist factor is sufficiently small then lambda is negative regardless ofall the other parameters in the system. But when all the parameters except theshear coefficient are fixed then lambda tends to infinity as the shearcoefficient tends to infinity.

Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise